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A derivative is the rate of change of a function with respect to an independent variable (usually time or just x). For example, the derivative of a position-versus-time graph (with the independent variable being time) is the rate of change of position as time passes. This is the velocity-versus-time graph. Similarly, the derivative of the velocity-versus-time graph is the acceleration-versus-time graph.
Here's another way to think about the derivative of a function. If you were to record the slopes of the tangent lines to each point on a graph, then put those values on a different graph, you would create a graph of the derivative of the original function. In other words, you would have found every instantaneous rate of change of the original function and graphed it in their own function.
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